Evolutes and isoperimetric deficit in two-dimensional spaces of constant curvature

Abstract

We relate the total curvature and the isoperimetric deficit of a curve $\gamma$ in a two-dimensional space of constant curvature with the area enclosed by the evolute of $\gamma$. We provide also a Gauss-Bonnet theorem for a special class of evolutes.Comment: 16 pages, 3 figure